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Theorem nosgnn0 31565
 Description: ∅ is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nosgnn0 ¬ ∅ ∈ {1𝑜, 2𝑜}

Proof of Theorem nosgnn0
StepHypRef Expression
1 1n0 7535 . . . 4 1𝑜 ≠ ∅
21nesymi 2847 . . 3 ¬ ∅ = 1𝑜
3 nsuceq0 5774 . . . . 5 suc 1𝑜 ≠ ∅
4 necom 2843 . . . . . 6 (suc 1𝑜 ≠ ∅ ↔ ∅ ≠ suc 1𝑜)
5 df-2o 7521 . . . . . . 7 2𝑜 = suc 1𝑜
65neeq2i 2855 . . . . . 6 (∅ ≠ 2𝑜 ↔ ∅ ≠ suc 1𝑜)
74, 6bitr4i 267 . . . . 5 (suc 1𝑜 ≠ ∅ ↔ ∅ ≠ 2𝑜)
83, 7mpbi 220 . . . 4 ∅ ≠ 2𝑜
98neii 2792 . . 3 ¬ ∅ = 2𝑜
102, 9pm3.2ni 898 . 2 ¬ (∅ = 1𝑜 ∨ ∅ = 2𝑜)
11 0ex 4760 . . 3 ∅ ∈ V
1211elpr 4176 . 2 (∅ ∈ {1𝑜, 2𝑜} ↔ (∅ = 1𝑜 ∨ ∅ = 2𝑜))
1310, 12mtbir 313 1 ¬ ∅ ∈ {1𝑜, 2𝑜}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 383   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∅c0 3897  {cpr 4157  suc csuc 5694  1𝑜c1o 7513  2𝑜c2o 7514 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3192  df-dif 3563  df-un 3565  df-nul 3898  df-sn 4156  df-pr 4158  df-suc 5698  df-1o 7520  df-2o 7521 This theorem is referenced by:  nosgnn0i  31566  sltres  31571  noseponlem  31575  sltso  31582  nodenselem3  31599  nodenselem8  31604
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